# Math Help - fermat...or not fermat?

1. ## fermat...or not fermat?

proove that if a prime number is represented by this phormula p=(2^n)+1 with n>0 then n is a power of 2... it's all for you...

2. Originally Posted by Aglaia
proove that if a prime number is represented by this phormula p=(2^n)+1 with n>0 then n is a power of 2... it's all for you...
The important fact is that, we can factor:
$x^n+y^n$ where $n$ is odd.

Now, assume,
$2^n+1$ is prime where $n$ is divisible by odd.
Then,
$2^{(2k+1)j}+1$
Thus,
$(2^j)^{2k+1}+1$
Can be factored as,
$(2^j+1)(2^{2kj}-2^{(2k-1)j}+2^{(2k-2)j}-...+2^{2j}-2^j+1)$
It has a proper nontrivial factorization, thus it cannot be prime.
Thus, $n$ cannot be divisble by odd number that is, $2^m$
Thus,
$2^{2^m}+1$
Which were studied by Fermat (my favorite mathemation).